What are some examples of functions that machine learning models compute?
If I understood correctly, the model is a polynomial equation

No, it's not true that all machine learning (ML) models compute (or represent) a polynomial function. For example, a sigmoid is not a polynomial, but, for example, in a neural network, you can combine many sigmoids to build complicated functions that may not necessarily be polynomials.
We usually distinguish between linear (straight-lines) and non-linear functions (rather than polynomials and non-polynomials). In some cases, it is straightforward to visualize the function that your model computes: for example, in the case of linear regression, once you learned the coefficients (i.e. the slope and y-intercept), you can plot the learned straight-line function. In other cases, for example, in the case of neural networks, it is not fully clear how to visualize the function that your model computes, given that it is the composition of many non-linear functions (typically, ReLUs, sigmoids or hyperbolic tangents).
If you are interested in solving problems with polynomials, take a look at polynomial regression.

and its weights are calculated by training the model with data sets.

Yes, in machine learning, we want to find a function that &quot;fits the given data&quot;, and the specific meaning of &quot;fitting the data&quot; depends on the specific machine learning technique.
For simplicity, let's focus on supervised learning, a machine learning technique where we are given a labelled dataset, i.e. a dataset of pairs $D = \{(x_1, y_1), \dots, (x_N, y_N)\}$, where we assume that $f(x_i) = y_i$, for some typically unknown function $f$, and $y_i$s are the labels (the outputs of $f$) and $x_i$ the inputs of $f$. The goal is to find function $g_{\theta}$ that approximates well $f$. I will soon describe what the subscript $\theta$ represents.
For simplicity, let's assume that $f$ is a linear function (i.e. a straight-line). So, we can define a linear model $g_{\theta}$ that we can use to find a function that approximates well $f$. Here is the linear model
$$g_{\theta}(x) = ax + b,$$
where

$g_{\theta}(x)$ is the output
$x$ is the input
$ax + b$ is the linear function (a straight-line)
$a$ is the slope (a parameter, aka weight)
$b$ is the $y$-intercept (another parameter)
$\theta = \{ a, b \}$ (the set of parameters of the linear model)

Why is this a model? I call this a model because, depending on the specific values of the parameters $\theta$, we have different specific functions. So, I am using the term &quot;model&quot; as a synonym for a set of functions, which, in this case, are limited by the definition $ax + b$ and the specific values that $a$ and $b$ (i.e. $\theta$) can take.
So, what do we do with this linear model? We want to find a specific set of parameters $\hat{\theta}$ (note that I use the $\hat{ }$ to emphasize that this is a specific configuration of the variable $\theta$) that corresponds to a linear function (a straight-line) that approximates $f$ well.  In other words, we need to find the parameters $\hat{\theta}$, such that $g_\hat{\theta} \approx f$, where $\approx$ means &quot;approximately computes&quot;.
How do we do that? We typically don't know $f$, but we know (or assumed) that $f(x_i) = y_i$, so the labeled dataset $D$ contains information about our unknown function $f$. So, the idea is that we can use the dataset $D$ to find a specific set of parameters $\hat{\theta}$ that corresponds to some function that approximates $f$ according to the information in $D$.
This process of finding $\hat{\theta}$ based on $D$ is often denoted as &quot;fitting the model to the data&quot;. There different ways of fitting the model to the data that differ in the way they compute some notion of distance between the information in $D$ and $g_{\hat{\theta}}$. I will not explain them here because this answer is already quite long. If you want to know more about it, you should take a book about the topic and read it.

What are some examples of functions that machine learning models compute?

I don't have specific examples, but you can easily try to fit a linear regression model to some labelled data, then plot the function that you found. You could use the Python library sklearn to do that.